Let ${\bf C}$ be a category, $\alpha$ an ordinal number, and $\zeta$ a function $\alpha \times \alpha \to \alpha$ (N.B.: ordinals have nothing special here, but since I'm working in TG, there is no loss of generality in using ordinals as indexing sets). Let us define a $5$-tuple $\zeta \ast {\bf C} := (D_o,D_h,u,v,d)$ as follows: $D_o$ is just the object class of $\bf C$, while $D_h$ is the quotient of $\coprod_{i \in \alpha} \hom({\bf C})$ by the equivalence $\mathcal R$ that amalgamates two couples $(f,i)$ and $(g,j)$ iff $f=g={\rm id}_{\bf C}(A)$ for some $A \in {\rm Ob}({\bf C})$; $u$ and $v$ are functions $D_h \to D_o$ taking the equivalence class $(f,i) \bmod \mathcal R$ to $s(f)$ and $t(f)$, respectively; and $d$ is the mapping $D_h \times_{v,u} D_h \to D_h$ taking a pair $((f,i)\bmod \mathcal R,(g,j) \bmod \mathcal R)$ to $(f,i) \bmod \mathcal R$, respectively to $(g,j) \bmod \mathcal R$, if $g$, respectively $f$, is an identity of $\bf C$, and to $(fg,\zeta(i,j)) \bmod \mathcal R$ otherwise.

It is clear that $\zeta \ast \bf C$ is itself a category: I'm using it as a toy example, while looking for something more "natural", to introduce a generalized notion of limit (and its dual) which, on the one hand, makes sense in much more abstract settings than categories (e.g., Ehresmann's neocats or Mitchell's semicats), and on the other hand, may perhaps be an effective surrogate of limits - I can't say, at present, if or not this is really the case - in situations where limits do not exist and weak limits are likely to not represent, for some reason, a satisfactory alternative. So, here are my questions:

Q1. Is $\zeta \ast \bf C$, up to equivalence, a disguised form of anything familiar to you? Q2. Is it maybe a particular colimit? Q3. Does it satisfy any "obvious" universal property?

Thanks in advance, as always, for any hint.